In algebra, a **constant** is a number whose value always remains the same and does not change. Constants stand in contrast to variables, which represent quantities that can take on different values. For example, if two people always differ in age by exactly 3 years, the number 3 is a constant—it stays fixed regardless of how old either person actually is. Constants are the fixed numerical components that appear alongside variables in algebraic expressions.

Algebraic Constant

Algebra uses specific symbols for the four fundamental arithmetic operations, each paired with standard terminology:

- **Addition**: Written as $$a + b$$, read "$$a$$ plus $$b$$." The result is called the **sum** of $$a$$ and $$b$$.
- **Subtraction**: Written as $$a - b$$, read "$$a$$ minus $$b$$." The result is called the **difference** of $$a$$ and $$b$$.
- **Multiplication**: Can be written as $$a \cdot b$$, $$(a)(b)$$, $$(a)b$$, $$a(b)$$, or simply $$ab$$, read "$$a$$ times $$b$$." The result is called the **product** of $$a$$ and $$b$$.
- **Division**: Can be written as $$a \div b$$, $$a/b$$, $$\frac{a}{b}$$, or using the long division symbol $$b\overline{)a}$$, read "$$a$$ divided by $$b$$." The result is called the **quotient** of $$a$$ and $$b$$; $$a$$ is the **dividend** and $$b$$ is the **divisor**.

When converting between English and symbols, the prepositions "of" and "and" are important cues. For instance, "the product of 4 and 8" means $$4 \cdot 8$$, and "the difference of 9 and 2" means $$9 - 2$$.

Arithmetic Operation Symbols in Algebra

The **equal sign** ($$=$$) is the symbol used in algebra to indicate that two quantities have the same value. The statement $$a = b$$ is read as "$$a$$ is equal to $$b$$" and asserts that whatever value $$a$$ represents is identical to the value represented by $$b$$. The equal sign serves as a fundamental connector in algebra, linking two expressions that are equivalent in value.

Equality Symbol in Algebra

A constant is a number whose value always stays the same.

Mathematical Constant

To write algebraically, we use operation symbols along with numbers and variables. The four basic arithmetic operations and their common notations are:
- Addition: $$a + b$$ represents the sum of $$a$$ and $$b$$.
- Subtraction: $$a - b$$ represents the difference of $$a$$ and $$b$$.
- Multiplication: $$a \cdot b$$, $$ab$$, $$(a)(b)$$, $$(a)b$$, or $$a(b)$$ represents the product of $$a$$ and $$b$$.
- Division: $$a \div b$$, $$a/b$$, or $$\frac{a}{b}$$ represents the quotient of $$a$$ and $$b$$.

Arithmetic Operation Symbols

A company's travel policy uses the formula $R = 0.60x$ to calculate mileage reimbursement for employees, where '$x$' represents the number of miles driven. Based on your understanding of algebraic terms, define what a 'variable' is and explain why the letter '$x$' is classified as a variable in this reimbursement formula.

Defining Variables in Corporate Expense Formulas

What does a variable such as $$x$$ represent in algebra?

In $$3x$$, what does $$x$$ represent?

What is the letter $$x$$ called when it represents a number that may change?

Match each algebra term with its meaning.

Question:
In the expression $$x+y$$, what are the letters $$x$$ and $$y$$ called? Define this algebraic term.

Sample answer:
The letters $$x$$ and $$y$$ are variables. A variable is a letter that represents a number whose value may change or be unknown.

Key points:
- $$x$$ and $$y$$ are variables.
- A variable is a letter that represents a number.
- The number may change or be unknown.

Feedback:
Identify the term variable and explain that it represents a number that may change or be unknown.

In the expression $$x+y$$, what are the letters $$x$$ and $$y$$ called? Define this algebraic term.

In algebra, a letter that represents an unknown or changing number is a _____.

True or false: In algebra, $$n$$ can be a variable that represents a changing or unknown number.

Put the steps for using a variable to represent an unknown number in order.

Case context:
The letter $$t$$ represents a number whose value is unknown or may change.

Question:
In the expression $$t+4$$, what algebraic term describes $$t$$? Define the term.

Sample answer:
The letter $$t$$ is a variable. A variable is a letter that represents a number whose value is unknown or may change.

Key points:
- $$t$$ is a variable.
- A variable is a letter that represents a number.
- The number may be unknown or may change.

Rubric:
Identify $$t$$ as a variable and explain that a variable represents a number whose value is unknown or may change.

In the expression $$t+4$$, what algebraic term describes $$t$$? Define the term.

In algebra, a variable is a letter of the alphabet used to represent a number whose value may change or is currently unknown. Commonly used symbols for variables include $$a$$, $$b$$, $$c$$, $$m$$, $$n$$, $$x$$, and $$y$$.

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Algebra

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OpenStax Intermediate Algebra

Mastering arithmetic operations—including those involving whole numbers, integers, fractions, decimals, and real numbers—provides the essential groundwork for studying elementary algebra. Just as a building depends on a carefully constructed foundation to remain structurally sound, a solid grasp of basic arithmetic along with an introduction to algebraic notation and vocabulary enables students to progress successfully to more advanced algebraic concepts and techniques.

Arithmetic as a Foundation for Algebra

**Prealgebra** is a branch of mathematics that serves as a bridge between arithmetic and elementary algebra. It reinforces mastery of basic operations with whole numbers, integers,
  fractions, and decimals while introducing the fundamental language of algebra, such as variables, expressions, and the solving of simple equations. By transitioning from
  concrete numerical calculations to abstract symbolic manipulation, Prealgebra provides the essential foundation for more advanced mathematical study.

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